Skip to main content
SpringerLink
Log in

Inequalities for One-Step Products

  • Conference paper
  • First Online:
Developments in Language Theory (DLT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11088))

Included in the following conference series:

  • 653 Accesses

Abstract

Let a be a letter of an alphabet A. Given a lattice of languages \(\mathcal {L}\), we describe the set of ultrafilter inequalities satisfied by the lattice \(\mathcal {L}_{a}\) generated by the languages of the form L or \(LaA^*\), where L is a language of \(\mathcal {L}\). We also describe the ultrafilter inequalities satisfied by the lattice \(\mathcal {L}_1\) generated by the lattices \(\mathcal {L}_{a}\), for \(a \in A\). When \(\mathcal {L}\) is a lattice of regular languages, we first describe the profinite inequalities satisfied by \(\mathcal {L}_a\) and \(\mathcal {L}_1\) and then provide a small basis of inequalities defining \(\mathcal {L}_1\) when \(\mathcal {L}\) is a Boolean algebra of regular languages closed under quotient.

The first author received financial support of FCT, through project UID/MULTI/04621/2013 of CEMAT-Ciências. The second author received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 670624) and is supported by the DeLTA project (ANR-16-CE40-0007).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Contrary to a frequent convention, we do include the empty set in \(\mathcal {C}(X)\).

  2. 2.

    This is actually a variation on some general results of [17] and [33, Theorem 1]. See also [7] for a nice survey.

References

  1. Almeida, J.: Residually finite congruences and quasi-regular subsets in uniform algebras. Portugaliæ Mathematica 46, 313–328 (1989)

    MathSciNet  MATH  Google Scholar 

  2. Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific Publishing Co. Inc., River Edge (1994). Translated from the 1992 Portuguese original and revised by the author

    Google Scholar 

  3. Almeida, J., Costa, A.: Infinite-vertex free profinite semigroupoids and symbolic dynamics. J. Pure Appl. Algebra 213(5), 605–631 (2009)

    Article  MathSciNet  Google Scholar 

  4. Almeida, J., Weil, P.: Profinite categories and semidirect products. J. Pure Appl. Algebra 123(1–3), 1–50 (1998)

    Article  MathSciNet  Google Scholar 

  5. Beer, G.: Topologies on Closed and Closed Convex Sets, Mathematics and its Applications, vol. 268. Kluwer Academic Publishers Group, Dordrecht (1993)

    Book  Google Scholar 

  6. Berge, C.: Espaces Topologiques: Fonctions Multivoques, Collection Universitaire de Mathématiques, vol. III. Dunod, Paris (1959)

    MATH  Google Scholar 

  7. Bonsangue, M.M., Kok, J.N.: Relating multifunctions and predicate transformers through closure operators. In: Hagiya, M., Mitchell, J.C. (eds.) TACS 1994. LNCS, vol. 789, pp. 822–843. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-57887-0_127

    Chapter  Google Scholar 

  8. Branco, M.J.J., Pin, J.-É.: Equations defining the polynomial closure of a lattice of regular languages. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 115–126. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02930-1_10

    Chapter  MATH  Google Scholar 

  9. Brzozowski, J.A., Čulík II, K., Gabrielian, A.: Classification of noncounting events. J. Comput. Syst. Sci. 5, 41–53 (1971)

    Article  MathSciNet  Google Scholar 

  10. Cohen, J., Perrin, D., Pin, J.-É.: On the expressive power of temporal logic for finite words. J. Comput. Syst. Sci. 46, 271–294 (1993)

    Article  Google Scholar 

  11. Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic Press, New York (1976)

    MATH  Google Scholar 

  12. Ésik, Z.: Extended temporal logic on finite words and wreath product of monoids with distinguished generators. In: Ito, M., Toyama, M. (eds.) DLT 2002. LNCS, vol. 2450, pp. 43–58. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-45005-X_4

    Chapter  Google Scholar 

  13. Gehrke, M., Grigorieff, S., Pin, J.-É.: Duality and equational theory of regular languages. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5126, pp. 246–257. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70583-3_21

    Chapter  MATH  Google Scholar 

  14. Gehrke, M., Grigorieff, S., Pin, J.-É.: A topological approach to recognition. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 151–162. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14162-1_13

    Chapter  Google Scholar 

  15. Gehrke, M., Krebs, A., Pin, J.-É.: Ultrafilters on words for a fragment of logic. Theor. Comput. Sci. 610(Part A), 37–58 (2016)

    Article  MathSciNet  Google Scholar 

  16. Gehrke, M., Petrişan, D., Reggio, L.: The Schützenberger product for syntactic spaces. In: Leibniz International Proceedings Information 43rd International Colloquium on Automata, Languages, and Programming, vol. 55, p. 112, Article no. 14. LIPIcs, Schloss Dagstuhl, Leibniz-Zent, Information, Wadern (2016)

    Google Scholar 

  17. Michael, E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71, 152–182 (1951)

    Article  MathSciNet  Google Scholar 

  18. Pin, J.-É.: Variétés de langages et variétés de semigroupes, thèse d’état. Université Paris VI (1981)

    Google Scholar 

  19. Pin, J.-É.: A variety theorem without complementation. Russ. Math. (Izvestija vuzov. Matematika) 39, 80–90 (1995)

    Google Scholar 

  20. Pin, J.-É.: Profinite methods in automata theory. In: Albers, S., Marion, J.-Y. (eds.) 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009), pp. 31–50, Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2009)

    Google Scholar 

  21. Pin, J.-É.: Equational descriptions of languages. Int. J. Found. Comput. Sci. 23, 1227–1240 (2012)

    Article  MathSciNet  Google Scholar 

  22. Pin, J.-É.: Dual space of a lattice as the completion of a pervin space. In: Höfner, P., Pous, D., Struth, G. (eds.) RAMICS 2017. LNCS, vol. 10226, pp. 24–40. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57418-9_2

    Chapter  Google Scholar 

  23. Pin, J.-É., Pinguet, A., Weil, P.: Ordered categories and ordered semigroups. Commun. Algebra 30, 5651–5675 (2002)

    Article  MathSciNet  Google Scholar 

  24. Pin, J.-É., Silva, P.V.: A topological approach to transductions. Theor. Comput. Sci. 340, 443–456 (2005)

    Article  MathSciNet  Google Scholar 

  25. Pin, J.-É., Straubing, H.: Some results on \(\cal{C}\)-varieties. Theor. Inform. Appl. 39, 239–262 (2005)

    Article  MathSciNet  Google Scholar 

  26. Pin, J.-É., Weil, P.: Profinite semigroups, Mal’cev products and identities. J. Algebra 182, 604–626 (1996)

    Article  MathSciNet  Google Scholar 

  27. Pin, J.-É., Weil, P.: Polynomial closure and unambiguous product. Theory Comput. Syst. 30, 1–39 (1997)

    Article  MathSciNet  Google Scholar 

  28. Pin, J.-É., Weil, P.: The wreath product principle for ordered semigroups. Commun. Algebra 30, 5677–5713 (2002)

    Article  MathSciNet  Google Scholar 

  29. Reiterman, J.: The Birkhoff theorem for finite algebras. Algebra Univ. 14(1), 1–10 (1982)

    Article  MathSciNet  Google Scholar 

  30. Rhodes, J., Steinberg, B.: The \(q\)-Theory of Finite Semigroups. Springer Monographs in Mathematics. Springer, New York (2009). https://doi.org/10.1007/b104443

    Book  MATH  Google Scholar 

  31. Rhodes, J., Tilson, B.: The kernel of monoid morphisms. J. Pure Appl. Algebra 62(3), 227–268 (1989)

    Article  MathSciNet  Google Scholar 

  32. Schützenberger, M.-P.: On finite monoids having only trivial subgroups. Inform. Control 8, 190–194 (1965)

    Article  MathSciNet  Google Scholar 

  33. Smyth, M.B.: Power domains and predicate transformers: a topological view. In: Diaz, J. (ed.) ICALP 1983. LNCS, vol. 154, pp. 662–675. Springer, Heidelberg (1983). https://doi.org/10.1007/BFb0036946

    Chapter  Google Scholar 

  34. Steinberg, B., Tilson, B.: Categories as algebra, II. Int. J. Algebra Comput. 13(6), 627–703 (2003)

    Article  MathSciNet  Google Scholar 

  35. Straubing, H.: Aperiodic homomorphisms and the concatenation product of recognizable sets. J. Pure Appl. Algebra 15(3), 319–327 (1979)

    Article  MathSciNet  Google Scholar 

  36. Straubing, H.: On logical descriptions of regular languages. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 528–538. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45995-2_46

    Chapter  Google Scholar 

  37. Thérien, D.: A language theoretic interpretation of the Schützenberger representations with applications to certain varieties of languages. Semigroup Forum 28(1–3), 235–248 (1984)

    Article  MathSciNet  Google Scholar 

  38. Tilson, B.: Categories as algebra: an essential ingredient in the theory of monoids. J. Pure Appl. Algebra 48(1–2), 83–198 (1987)

    Article  MathSciNet  Google Scholar 

  39. Weil, P.: Products of languages with counter. Theor. Comput. Sci. 76, 251–260 (1990)

    Article  MathSciNet  Google Scholar 

  40. Weil, P.: Closure of varieties of languages under products with counter. J. Comput. Syst. Sci. 45, 316–339 (1992)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEMAT-Ciências and Dep. Matemática da Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016, Lisboa, Portugal

    Mário J. J. Branco

  2. IRIF, Université Paris 7 and CNRS, Case 7014, 75205, Paris Cedex 13, France

    Jean-Éric Pin

Authors
  1. Mário J. J. Branco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Éric Pin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Éric Pin .

Editor information

Editors and Affiliations

  1. National Archives of Japan, Tokyo, Japan

    Mizuho Hoshi

  2. The University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Branco, M.J.J., Pin, JÉ. (2018). Inequalities for One-Step Products. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-98654-8_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-98653-1

  • Online ISBN: 978-3-319-98654-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation